Chapter II: Part 2

¶ Her{e} he puttes þe thryde case of þe craft of Addicioɳ. & þe case is þis. yf of Addiciouɳ of 2 figuris a-ryse an Articull{e}, how schal þ{o}u do. thou most do away þe heer fig{ur}e þ{a}t was addid to þe neþ{er}, & write þ{ere} a cifre, and sett þe articuls on þe figuris hede, yf þ{a}t þ{ere} come ony aft{er}. And wyrch þan as I haue tolde þe in þe secunde case. An ensampull.

25.
15

Cast 5 to 5, þat wylle be ten. now do away þe hyer 5, & write þ{ere} a cifer. And sette ten vpon þe figuris hed of 2. And reken it but for on þus.] lo an Ensampull{e}

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And [*leaf 142a] þan worch forth. But yf þ{ere} come no figure aft{er} þe cifre, write þe articul next hym in þe same rewe as here

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cast 5 to 5, and it wel be ten. do away 5. þat is þe hier 5. and write þ{ere} a cifre, & write aft{er} hym þe articul as þus

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| 5 |
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And þan þ{o}u hast done.

¶ Si tibi cifra sup{er}ueniens occurrerit, illa{m}
Dele sup{er}posita{m}; fac illic scribe figura{m},
Postea procedas reliquas addendo figuras.

[Sidenote: What to do when you have a cipher in the top row.
An example of all the difficulties.]

¶ Her{e} he putt{es} þe fourt case, & it is þis, þat yf þ{ere} come a cifer in þe hier rewe, how þ{o}u schal do. þus þ{o}u schalt do. do away þe cifer, & sett þ{ere} þe digit þ{a}t comes of þe addiciou{n} as þus

1φφ84.
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In þis ensampul ben all{e} þe four{e} cases. Cast 3 to foure, þ{a}t wol be seueɳ. do away 4. & write þ{ere} seueɳ; þan cast 4 to þe figur{e} of 8. þ{a}t wel be 12. do away 8, & sett þ{ere} 2. þat is a digit, and sette þe articul of þe composit, þat is ten, vpon þe cifers hed, & reken it for hym selfe þat is on. þan cast on{e} to a cifer, & hit wull{e} be but on, for noȝt & on makes but on{e}. þan cast 7. þ{a}t stondes vnd{er} þat on to hym, & þat wel be 8. do away þe cifer & þat 1. & sette þ{ere} 8. þan go forthermor{e}. cast þe oþ{er} 7 to þe cifer þ{a}t stondes ou{er} hy{m}. þ{a}t wul be bot seuen, for þe cifer betokens noȝt. do away þe cifer & sette þ{ere} seueɳ, [*leaf 142b] & þen go forþ{er}mor{e} & cast 1 to 1, & þat wel be 2. do away þe hier 1, & sette þ{ere} 2. þan hast þ{o}u do. And yf þ{o}u haue wel ydo þis nomber þat is sett her{e}-aft{er} wel be þe nomber þat schall{e} aryse of all{e} þe addicioɳ as her{e} 27827. ¶ Sequi{tu}r alia sp{eci}es.

[Headnote: The Craft of Subtraction.]

++A nu{mer}o num{er}u{m} si sit tibi demer{e} cura
Scribe figurar{um} series, vt in addicione.

[Sidenote: Four things to know about subtraction: the first;
the second; the third; the fourth.]

¶ This is þe Chapt{er} of subtraccioɳ, in the quych þou most know foure nessessary thyng{es}. the first what is subtraccioɳ. þe secunde is how mony nombers þou most haue to subt{ra}ccioɳ, the thryd is how mony maners of cases þ{ere} may happe in þis craft of subtraccioɳ. The fourte is qwat is þe p{ro}fet of þis craft. ¶ As for þe first, þ{o}u most know þ{a}t subtraccioɳ is drawyng{e} of on{e} nowmb{er} oute of anoþ{er} nomber. As for þe secunde, þou most knowe þ{a}t þou most haue two rewes of figuris on{e} vnd{er} anoþ{er}, as þ{o}u addyst in addicioɳ. As for þe thryd, þ{o}u moyst know þ{a}t four{e} man{er} of diu{er}se casis mai happe in þis craft. ¶ As for þe fourt, þou most know þ{a}t þe p{ro}fet of þis craft is whenne þ{o}u hasse taken þe lasse nomber out of þe mor{e} to telle what þ{ere} leues ou{er} þ{a}t. & þ{o}u most be-gynne to wyrch in þ{is} craft in þe ryght side of þe boke, as þ{o}u diddyst in addicioɳ. V{er}sus.

¶ Maiori nu{mer}o num{er}u{m} suppone minorem,
¶ Siue pari nu{mer}o supponat{ur} num{er}us par.

[Sidenote: Put the greater number above the less.]

[*leaf 143a] ¶ Her{e} he telles þat þe hier nomber most be mor{e} þen þe neþ{er}, or els eueɳ as mych. but he may not be lasse. And þe case is þis, þou schalt drawe þe neþ{er} nomber out of þe hyer, & þou mayst not do þ{a}t yf þe hier nomber wer{e} lasse þan þat. ffor þ{o}u mayst not draw sex out of 2. But þ{o}u mast draw 2 out of sex. And þou maiste draw twene out of twene, for þou schal leue noȝt of þe hier twene vn{de} v{er}sus.

[Headnote: The Cases of the Craft of Subtraction.]

¶ Postea si possis a prima subt{ra}he p{ri}ma{m}
Scribens quod remanet.

[Sidenote: The first case of subtraction. Here is an example.]

Her{e} is þe first case put of subtraccioɳ, & he says þou schalt begynne in þe ryght side, & draw þe first fig{ure} of þe neþ{er} rewe out of þe first fig{ure} of þe hier rewe. qwether þe hier fig{ur}e be mor{e} þen þe neþ{er}, or eueɳ as mych. And þat is notified in þe vers when he says “Si possis.” Whan þ{o}u has þus ydo, do away þe hiest fig{ur}e & sett þ{ere} þat leues of þe subtraccioɳ, lo an Ensampull{e}

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draw 2 out of 4. þan leues 2. do away 4 & write þ{ere} 2, & latte þe neþ{er} figur{e} sto{n}de stille, & so go for-by oþ{er} figuris till þ{o}u come to þe ende, þan hast þ{o}u do.

¶ Cifram si nil remanebit.

[Sidenote: Put a cipher if nothing remains. Here is an example.]

¶ Her{e} he putt{es} þe secunde case, & hit is þis. yf it happe þ{a}t qwen þ{o}u hast draw on neþ{er} fig{ure} out of a hier, & þ{er}e leue noȝt aft{er} þe subt{ra}ccioɳ, þus [*leaf 143b] þou schalt do. þ{o}u schall{e} do away þe hier fig{ur}e & write þ{ere} a cifer, as lo an Ensampull

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Take four{e} out of four{e} þan leus noȝt. þ{er}efor{e} do away þe hier 4 & set þ{ere} a cifer, þan take 2 out of 2, þan leues noȝt. do away þe hier 2, & set þ{ere} a cifer, and so worch whar{e} so eu{er} þis happe.

Sed si no{n} possis a p{ri}ma dem{er}e p{ri}ma{m}
P{re}cedens vnu{m} de limite deme seque{n}te,
Quod demptu{m} p{ro} denario reputabis ab illo
Subt{ra}he to{ta}lem num{er}u{m} qu{em} p{ro}posuisti
Quo facto sc{ri}be super quicquid remaneb{i}t.

[Sidenote: Suppose you cannot take the lower figure from the top one,
borrow ten; take the lower number from ten; add the answer to the top
number. How to ‘Pay back’ the borrowed ten. Example.]

Her{e} he puttes þe thryd case, þe quych is þis. yf it happe þat þe neþ{er} fig{ur}e be mor{e} þen þe hier fig{ur}e þat he schall{e} be draw out of. how schall{e} þou do. þus þ{o}u schall{e} do. þou schall{e} borro .1. oute of þe next fig{ur}e þat comes aft{er} in þe same rewe, for þis case may neu{er} happ but yf þ{ere} come figures aft{er}. þan þ{o}u schalt sett þat on ou{er} þe hier figur{es} hed, of the quych þou woldist y-draw oute þe neyþ{er} fig{ur}e yf þ{o}u haddyst y-myȝt. Whane þou hase þus ydo þou schall{e} rekene þ{a}t .1. for ten. ¶. And out of þat ten þ{o}u schal draw þe neyþermost fig{ur}e, And all{e} þ{a}t leues þou schall{e} adde to þe figur{e} on whos hed þat .1. stode. And þen þ{o}u schall{e} do away all{e} þat, & sett þ{ere} all{e} that arisys of the addicioɳ of þe ylke 2 fig{ur}is. And yf yt [*leaf 144a] happe þat þe fig{ur}e of þe quych þ{o}u schalt borro on be hym self but 1. If þ{o}u schalt þat on{e} & sett it vppoɳ þe oþ{er} figur{is} hed, and sett in þ{a}t 1. place a cifer, yf þ{ere} come mony figur{es} aft{er}. lo an Ensampul.

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take 4 out of 2. it wyl not be, þerfor{e} borro on{e} of þe next figur{e}, þ{a}t is 2. and sett þat ou{er} þe hed of þe fyrst 2. & rekene it for ten. and þere þe secunde stondes write 1. for þ{o}u tokest on out of hy{m}. þan take þe neþ{er} fig{ur}e, þat is 4, out of ten. And þen leues 6. cast to 6 þe fig{ur}e of þat 2 þat stode vnd{er} þe hedde of 1. þat was borwed & rekened for ten, and þat wylle be 8. do away þ{a}t 6 & þat 2, & sette þ{ere} 8, & lette þe neþ{er} fig{ur}e stonde stille. Whanne þ{o}u hast do þus, go to þe next fig{ur}e þ{a}t is now bot 1. but first yt was 2, & þ{ere}-of was borred 1. þan take out of þ{a}t þe fig{ur}e vnd{er} hym, þ{a}t is 3. hit wel not be. þer-for{e} borowe of the next fig{ur}e, þe quych is bot 1. Also take & sett hym ou{er} þe hede of þe fig{ure} þat þou woldest haue y-draw oute of þe nether figure, þe quych was 3. & þou myȝt not, & rekene þ{a}t borwed 1 for ten & sett in þe same place, of þe quych place þ{o}u tokest hy{m} of, a cifer, for he was bot 1. Whanne þ{o}u hast þ{us} ydo, take out of þat 1. þ{a}t is rekent for ten, þe neþ{er} figure of 3. And þ{ere} leues 7. [*leaf 144b] cast þe ylke 7 to þe fig{ur}e þat had þe ylke ten vpon his hed, þe quych fig{ur}e was 1, & þat wol be 8. þan do away þ{a}t 1 and þ{a}t 7, & write þ{ere} 8. & þan wyrch forth in oþ{er} figuris til þ{o}u come to þe ende, & þan þ{o}u hast þe do. V{er}sus.

¶ Facque nonenarios de cifris, cu{m} remeabis
¶ Occ{ur}rant si forte cifre; dum demps{er}is vnum
¶ Postea p{ro}cedas reliquas deme{n}do figuras.

[Sidenote: A very hard case is put. Here is an example.]

¶ Her{e} he putt{es} þe fourte case, þe quych is þis, yf it happe þat þe neþ{er} fig{ur}e, þe quych þ{o}u schalt draw out of þe hier fig{ur}e be mor{e} pan þe hier figur ou{er} hym, & þe next fig{ur}e of two or of thre or of foure, or how mony þ{ere} be by cifers, how wold þ{o}u do. Þ{o}u wost wel þ{o}u most nede borow, & þ{o}u mayst not borow of þe cifers, for þai haue noȝt þat þai may lene or spar{e}. Ergo[{4}] how woldest þ{o}u do. Certayɳ þus most þ{o}u do, þ{o}u most borow on of þe next figure significatyf in þat rewe, for þis case may not happe, but yf þ{ere} come figures significatyf aft{er} the cifers. Whan þ{o}u hast borowede þ{a}t 1 of the next figure significatyf, sett þ{a}t on ou{er} þe hede of þ{a}t fig{ur}e of þe quych þ{o}u wold haue draw þe neþ{er} figure out yf þ{o}u hadest myȝt, & reken it for ten as þo{u} diddest i{n} þe oþ{er} case her{e}-a-for{e}. Whaɳ þ{o}u hast þus y-do loke how mony cifers þ{ere} wer{e} bye-twene þat figur{e} significatyf, & þe fig{ur}e of þe quych þ{o}u woldest haue y-draw the [*leaf 145a] neþ{er} figure, and of eu{er}y of þe ylke cifers make a figur{e} of 9. lo an Ensampull{e} after.

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Take 4 out of 2. it wel not be. borow 1 out of be next figure significatyf, þe quych is 4, & þen leues 3. do away þ{a}t figur{e} of 4 & write þ{ere} 3. & sett þ{a}t 1 vppon þe fig{ur}e of 2 hede, & þan take 4 out of ten, & þan þere leues 6. Cast 6 to the fig{ur}e of 2, þ{a}t wol be 8. do away þat 6 & write þ{er}e 8. Whan þ{o}u hast þus y-do make of eu{er}y 0 betweyn 3 & 8 a figure of 9, & þan worch forth in goddes name. & yf þ{o}u hast wel y-do þ{o}u[{5}] schalt haue þis nomb{er}

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|39998| Sic.
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[Headnote: How to prove the Subtraction.]

¶ Si subt{ra}cc{i}o sit b{e}n{e} facta p{ro}bar{e} valebis
Quas s{u}btraxisti p{ri}mas addendo figuras.

[Sidenote: How to prove a subtraction sum. Here is an example.
He works his proof through, and brings out a result.]

¶ Her{e} he teches þe Craft how þ{o}u schalt know, whan þ{o}u hast subt{ra}yd, wheþ{er} þou hast wel ydo or no. And þe Craft is þis, ryght as þ{o}u subtrayd þe neþ{er} figures fro þe hier figures, ryȝt so adde þe same neþ{er} figures to þe hier figures. And yf þ{o}u haue well y-wroth a-for{e} þou schalt haue þe hier nombre þe same þ{o}u haddest or þou be-gan to worch. as for þis I bade þou schulde kepe þe neþ{er} figures stylle. lo an [*leaf 145b] Ensampull{e} of all{e} þe 4 cases toged{re}. worche well{e} þis case

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And yf þou worch well{e} whan þou hast all{e} subtrayd þe þ{a}t hier nombr{e} her{e}, þis schall{e} be þe nombre here foloyng whan þ{o}u hast subtrayd.

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|39998804|. [Sidenote: Our author makes a slip here (3 for 1).]
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And þou schalt know þ{us}. adde þe neþ{er} rowe of þe same nombre to þe hier rewe as þus, cast 4 to 4. þat wol be 8. do away þe 4 & write þ{ere} 8. by þe first case of addicioɳ. þan cast 6 to 0 þat wol be 6. do away þe 0, & write þere 6. þan cast 6 to 8, þ{a}t wel be 14. do away 8 & write þ{ere} a fig{ur}e of 4, þat is þe digit, and write a fig{ur}e of 1. þ{a}t schall be-token ten. þ{a}t is þe articul vpon þe hed of 8 next aft{er}, þan reken þ{a}t 1. for 1. & cast it to 8. þat schal be 9. cast to þat 9 þe neþ{er} fig{ur}e vnd{er} þat þe quych is 4, & þat schall{e} be 13. do away þat 9 & sett þ{er}e 3, & sett a figure of 1. þ{a}t schall be 10 vpon þe next figur{is} hede þe quych is 9. by þe secu{n}de case þ{a}t þ{o}u hadest in addicioɳ. þan cast 1 to 9. & þat wol be 10. do away þe 9. & þat 1. And write þ{ere} a cifer. and write þe articull{e} þat is 1. betokenyng{e} 10. vpon þe hede of þe next figur{e} toward þe lyft side, þe quych [*leaf 146a] is 9, & so do forth tyl þ{o}u come to þe last 9. take þe figur{e} of þat 1. þe quych þ{o}u schalt fynde ou{er} þe hed of 9. & sett it ou{er} þe next figures hede þat schal be 3. ¶ Also do away þe 9. & set þ{ere} a cifer, & þen cast þat 1 þat stondes vpon þe hede of 3 to þe same 3, & þ{a}t schall{e} make 4, þen caste to þe ylke 4 the figur{e} in þe neyþ{er} rewe, þe quych is 2, and þat schall{e} be 6. And þen schal þ{o}u haue an Ensampull{e} aȝeyɳ, loke & se, & but þ{o}u haue þis same þ{o}u hase myse-wroȝt.

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Sequit{ur} de duplac{i}one

[Headnote: The Craft of Duplation.]

++Si vis duplar{e} num{er}u{m}, sic i{n}cipe p{rim}o
Scribe fig{ur}ar{um} serie{m} q{ua}mcu{n}q{ue} vel{is} tu.

[Sidenote: Four things must be known in Duplation. Here they are.
Mind where you begin. Remember your rules.]

¶ This is the Chaptur{e} of duplacioɳ, in þe quych craft þ{o}u most haue & know 4 thing{es}. ¶ Þe first þ{a}t þ{o}u most know is what is duplacioɳ. þe secu{n}de is how mony rewes of fig{ur}es þ{o}u most haue to þis craft. ¶ þe thryde is how many cases may[{6}] happe in þis craft. ¶ þe fourte is what is þe p{ro}fet of þe craft. ¶ As for þe first. duplacioɳ is a doublyng{e} of a nombre. ¶ As for þe secu{n}de þ{o}u most [*leaf 146b] haue on nombre or on rewe of figures, the quych called nu{merus} dupland{us}. As for þe thrid þ{o}u most know þat 3 diu{er}se cases may hap in þis craft. As for þe fourte. qwat is þe p{ro}fet of þis craft, & þ{a}t is to know what a-risyȝt of a nombre I-doublyde. ¶ fforþ{er}-mor{e}, þ{o}u most know & take gode hede in quych side þ{o}u schall{e} be-gyn in þis craft, or ellis þ{o}u mayst spyl all{e} þ{i} lab{er} þ{er}e aboute. c{er}teyn þ{o}u schalt begyɳ in the lyft side in þis Craft. thenke wel ou{er} þis verse. ¶ [{7}]A leua dupla, diuide, m{u}ltiplica.[{7}] [[Subt{ra}has a{u}t addis a dext{ri}s {ve}l medi{a}b{is}]] The sentens of þes verses afor{e}, as þ{o}u may see if þ{o}u take hede. As þe text of þis verse, þat is to say, ¶ Si vis duplare. þis is þe sentence. ¶ If þ{o}u wel double a nombre þus þ{o}u most be-gynɳ. Write a rewe of figures of what nomb{re} þou welt. v{er}sus.

Postea p{ro}cedas p{ri}ma{m} duplando figura{m}
Inde q{uo}d excrescit scribas vbi iusserit ordo
Iuxta p{re}cepta tibi que dant{ur} in addic{i}one.

[Sidenote: How to work a sum.]

¶ Her{e} he telles how þ{o}u schalt worch in þis Craft. he says, fyrst, whan þ{o}u hast writen þe nombre þ{o}u schalt be-gyn at þe first figur{e} in the lyft side, & doubull{e} þat fig{ur}e, & þe nombre þat comes þ{ere}-of þ{o}u schalt write as þ{o}u diddyst in addicioɳ, as ¶ I schal telle þe in þe case. v{er}sus.

[Headnote: The Cases of the Craft of Duplation.]

[*leaf 147a]

¶ Nam si sit digitus in primo limite scribas.

[Sidenote: If the answer is a digit, write it in the place of the
top figure.]

¶ Her{e} is þe first case of þis craft, þe quych is þis. yf of duplacioɳ of a figur{e} arise a digit. what schal þ{o}u do. þus þ{o}u schal do. do away þe fig{ur}e þat was doublede, & sett þ{ere} þe diget þat comes of þe duplacioɳ, as þus. 23. double 2, & þ{a}t wel be 4. do away þe figur{e} of 2 & sett þ{ere} a figur{e} of 4, & so worch forth till{e} þ{o}u come to þe ende. v{er}sus.

¶ Articul{us} si sit, in p{ri}mo limite cifram,
¶ Articulu{m} v{er}o reliquis inscribe figuris;
¶ Vel p{er} se scribas, si nulla figura sequat{ur}.

[Sidenote: If it is an article, put a cipher in the place, and
‘carry’ the tens. If there is no figure to ‘carry’ them to, write
them down.]

¶ Here is þe secunde case, þe quych is þis yf þ{ere} come an articull{e} of þe duplacioɳ of a fig{ur}e þ{o}u schalt do ryȝt as þ{o}u diddyst in addicioɳ, þat is to wete þat þ{o}u schalt do away þe figur{e} þat is doublet & sett þ{ere} a cifer, & write þe articull{e} ou{er} þe next figur{is} hede, yf þ{ere} be any aft{er}-warde toward þe lyft side as þus. 25. begyn at the lyft side, and doubull{e} 2. þat wel be 4. do away þat 2 & sett þere 4. þan doubul 5. þat wel be 10. do away 5, & sett þ{ere} a 0, & sett 1 vpon þe next figur{is} hede þe quych is 4. & þen draw downe 1 to 4 & þat woll{e} be 5, & þen do away þ{a}t 4 & þat 1, & sett þ{ere} 5. for þat 1 schal be rekened in þe drawyng{e} toged{re} for 1. wen [*leaf 147b] þou hast ydon þou schalt haue þis nomb{r}e 50. yf þ{ere} come no figur{e} aft{er} þe fig{ur}e þ{a}t is addit, of þe quych addicioɳ comes an articull{e}, þ{o}u schalt do away þe figur{e} þ{a}t is dowblet & sett þ{ere} a 0. & write þe articul next by in þe same rewe toward þe lyft syde as þus, 523. double 5 þat woll be ten. do away þe figur{e} 5 & set þ{ere} a cifer, & sett þe articul next aft{er} in þe same rewe toward þe lyft side, & þou schalt haue þis nombre 1023. þen go forth & double þe oþ{er} nombers þe quych is lyȝt y-nowȝt to do. v{er}sus.

¶ Compositus si sit, in limite sc{ri}be seq{uen}te
Articulu{m}, p{ri}mo digitu{m}; q{uia} sic iubet ordo:
Et sic de reliq{ui}s facie{n}s, si sint tibi plures.

[Sidenote: If it is a Composite, write down the digit, and ‘carry’
the tens. Here is an example.]

¶ Her{e} he putt{es} þe Thryd case, þe quych is þis, yf of duplacioɳ of a fig{ur}e come a Composit. þ{o}u schalt do away þe fig{u}re þ{a}t is doublet & set þ{ere} a digit of þe Composit, & sett þe articull{e} ou{er} þe next figures hede, & aft{er} draw hym downe w{i}t{h} þe figur{e} ou{er} whos hede he stondes, & make þ{ere}-of an nombre as þ{o}u hast done afore, & yf þ{ere} come no fig{ur}e aft{er} þat digit þat þ{o}u hast y-write, þa{n} set þe articull{e} next aft{er} hym in þe same rewe as þus, 67: double 6 þat wel be 12, do away 6 & write þ{ere} þe digit [*leaf 148a] of 12, þe quych is 2, and set þe articull{e} next aft{er} toward þe lyft side in þe same rewe, for þ{ere} comes no figur{e} aft{er}. þan dowble þat oþ{er} figur{e}, þe quych is 7, þat wel be 14. the quych is a Composit. þen do away 7 þat þ{o}u doublet & sett þe þe diget of hy{m}, the quych is 4, sett þe articull{e} ou{er} þe next figur{es} hed, þe quych is 2, & þen draw to hym þat on, & make on nombre þe quych schall{e} be 3. And þen yf þ{o}u haue wel y-do þ{o}u schall{e} haue þis nombre of þe duplacioɳ, 134. v{er}sus.

¶ Si super ext{re}ma{m} nota sit monade{m} dat eid{em}
Quod t{ibi} {con}tingat si p{ri}mo dimidiabis.

[Sidenote: How to double the mark for one-half. This can only stand
over the first figure.]

¶ Her{e} he says, yf ou{er} þe fyrst fig{ur}e in þe ryȝt side be such a merke as is her{e} made, ʷ, þ{o}u schall{e} fyrst doubull{e} þe figur{e}, the quych stondes vnd{er} þ{a}t merke, & þen þou schalt doubul þat merke þe quych stond{es} for haluendel on. for too haluedels makes on, & so þ{a}t wol be on. cast þ{a}t on to þat duplacioɳ of þe figur{e} ou{er} whos hed stode þat merke, & write it in þe same place þ{ere} þat þe figur{e} þe quych was doublet stode, as þus 23ʷ. double 3, þat wol be 6; doubul þat halue on, & þat wol be on. cast on to 6, þ{a}t wel be 7. do away 6 & þat 1, & sett þ{ere} 7. þan hase þou do. as for þat figur{e}, þan go [*leaf 148b] to þe oþ{er} fig{ure} & worch forth. & þ{o}u schall neu{er} haue such a merk but ou{er} þe hed of þe furst figure in þe ryght side. And ȝet it schal not happe but yf it were y-halued a-for{e}, þus þ{o}u schalt vnd{er}stonde þe verse. ¶ Si sup{er} ext{re}ma{m} &c. Et nota, talis fig{ur}a ʷ significans medietate{m}, unitat{is} veniat, {i.e.} contingat u{e}l fiat sup{er} ext{re}ma{m}, {i.e.} sup{er} p{ri}ma{m} figura{m} in ext{re}mo sic v{er}sus dextram ars dat: {i.e.} reddit monade{m}. {i.e.} vnitate{m} eide{m}. {i.e.} eidem note & declina{tur} hec monos, d{i}s, di, dem, &c. ¶ Quod {er}g{o} to{tum} ho{c} dabis monade{m} note {con}ting{et}. {i.e.} eveniet tibi si dimidiasti, {i.e.} accipisti u{e}l subtulisti medietatem alicuius unius, in cuius principio sint figura nu{mer}u{m} denotans i{m}pare{m} p{rim}o {i.e.} principiis.

[Headnote: The Craft of Mediation.]

¶ Sequit{ur} de mediacione.

++Incipe sic, si vis alique{m} nu{me}ru{m} mediar{e}:
Sc{ri}be figurar{um} seriem sola{m}, velut an{te}.

[Sidenote: The four things to be known in mediation: the first the
second; the third; the fourth. Begin thus.]

¶ In þis Chapter is taȝt þe Craft of mediaciouɳ, in þe quych craft þ{o}u most know 4 thynges. ffurst what is mediacioɳ. the secunde how mony rewes of figur{es} þ{o}u most haue in þe wyrchyng{e} of þis craft. þe thryde how mony diu{er}se cases may happ in þis craft.[{8}] [[the .4. what is þe p{ro}fet of þis craft.]] ¶ As for þe furst, þ{o}u schalt vndurstonde þat mediacioɳ is a takyng out of halfe a nomber out of a holle nomber, [*leaf 149a] as yf þ{o}u wolde take 3 out of 6. ¶ As for þe secunde, þ{o}u schalt know þ{a}t þ{o}u most haue on{e} rewe of figures, & no moo, as þ{o}u hayst in þe craft of duplacioɳ. ¶ As for the thryd, þou most vnd{er}stonde þat 5 cases may happe in þis craft. ¶ As for þe fourte, þou schall{e} know þat the p{ro}fet of þis craft is when þ{o}u hast take away þe haluendel of a nomb{re} to telle qwat þer{e} schall{e} leue. ¶ Incipe sic, &c. The sentence of þis verse is þis. yf þ{o}u wold medye, þat is to say, take halfe out of þe holle, or halfe out of halfe, þou most begynne þ{us}. Write on{e} rewe of figur{es} of what nombre þou wolte, as þ{o}u dyddyst be-for{e} in þe Craft of duplacioɳ. v{er}sus.

¶ Postea p{ro}cedas medians, si p{ri}ma figura
Si par aut i{m}par videas.

[Sidenote: See if the number is even or odd.]

¶ Her{e} he says, when þ{o}u hast write a rewe of figures, þ{o}u schalt take hede wheþ{er} þe first figur{e} be eueɳ or odde in nombre, & vnd{er}stonde þ{a}t he spekes of þe first figure in þe ryȝt side. And i{n} the ryght side þ{o}u schall{e} begynne in þis Craft.

¶ Quia si fu{er}it par,
Dimidiab{is} eam, scribe{n}s quicq{ui}d remanebit:

[Sidenote: If it is even, halve it, and write the answer in its
place.]

¶ Her{e} is the first case of þis craft, þe quych is þis, yf þe first figur{e} be euen. þou schal take away fro þe figur{e} euen halfe, & do away þat fig{ur}e and set þ{ere} þat leues ou{er}, as þus, 4. take [*leaf 149b] halfe out of 4, & þan þ{ere} leues 2. do away 4 & sett þ{ere} 2. þis is lyght y-nowȝt. v{er}sus.

[Headnote: The Mediation of an Odd Number.]

¶ Impar si fu{er}it vnu{m} demas mediar{e}
Quod no{n} p{re}sumas, s{ed} quod sup{er}est mediabis
Inde sup{er} tractu{m} fac demptu{m} quod no{ta}t vnu{m}.

[Sidenote: If it is odd, halve the even number less than it. Here is
an example. Then write the sign for one-half over it. Put the mark
only over the first figure.]

Her{e} is þe secunde case of þis craft, the quych is þis. yf þe first figur{e} betoken{e} a nombre þat is odde, the quych odde schal not be mediete, þen þ{o}u schalt medye þat nombre þat leues, when the odde of þe same nomb{re} is take away, & write þat þ{a}t leues as þ{o}u diddest in þe first case of þis craft. Whaɳ þ{o}u hayst write þat. for þ{a}t þat leues, write such a merke as is her{e} ʷ vpon his hede, þe quych merke schal betokeɳ halfe of þe odde þat was take away. lo an Ensampull. 245. the first figur{e} her{e} is betokenyng{e} odde nombre, þe quych is 5, for 5 is odde; þ{er}e-for{e} do away þat þ{a}t is odde, þe quych is 1. þen leues 4. þen medye 4 & þen leues 2. do away 4. & sette þ{ere} 2, & make such a merke ʷ upon his hede, þat is to say ou{er} his hede of 2 as þus. 242.ʷ And þen worch forth in þe oþ{er} figures tyll þ{o}u come to þe ende. by þe furst case as þ{o}u schalt vnd{er}stonde þat þ{o}u schalt [*leaf 150a] neu{er} make such a merk but ou{er} þe first fig{ur}e hed in þe riȝt side. Wheþ{er} þe other fig{ur}es þat comyɳ aft{er} hym be eueɳ or odde. v{er}sus.

[Headnote: The Cases of the Craft of Mediation.]

¶ Si monos, dele; sit t{ibi} cifra post no{ta} supra.

[Sidenote: If the first figure is one put a cipher.]

¶ Here is þe thryde case, þe quych yf the first figur{e} be a figur{e} of 1. þ{o}u schalt do away þat 1 & set þ{ere} a cifer, & a merke ou{er} þe cifer as þus, 241. do away 1, & sett þ{ere} a cifer w{i}t{h} a merke ou{er} his hede, & þen hast þ{o}u ydo for þat 0. as þus 0ʷ þen worch forth in þe oþer fig{ur}ys till þ{o}u come to þe ende, for it is lyght as dyche water. vn{de} v{er}sus.

¶ Postea p{ro}cedas hac condic{i}one secu{n}da:
Imp{ar} si fu{er}it hinc vnu{m} deme p{ri}ori,
Inscribens quinque, nam denos significabit
Monos p{re}d{ict}am.

[Sidenote: What to do if any other figure is odd. Write a figure of
five over the next lower number’s head. Example.]

¶ Her{e} he putt{es} þe fourte case, þe quych is þis. yf it happeɳ the secunde figur{e} betoken odde nombre, þou schal do away on of þat odde nombre, þe quych is significatiue by þ{a}t figure 1. þe quych 1 schall be rekende for 10. Whan þ{o}u hast take away þ{a}t 1 out of þe nombre þ{a}t is signifiede by þat figur{e}, þ{o}u schalt medie þ{a}t þat leues ou{er}, & do away þat figur{e} þat is medied, & sette in his styde halfe of þ{a}t nombre. ¶ Whan þ{o}u hase so done, þ{o}u schalt write [*leaf 150b] a figure of 5 ou{er} þe next figur{es} hede by-for{e} toward þe ryȝt side, for þat 1, þe quych made odd nombr{e}, schall stonde for ten, & 5 is halfe of 10; so þ{o}u most write 5 for his haluendell{e}. lo an Ensampull{e}, 4678. begyɳ in þe ryȝt side as þ{o}u most nedes. medie 8. þen þ{o}u schalt leue 4. do away þat 8 & sette þ{ere} 4. þen out of 7. take away 1. þe quych makes odde, & sett 5. vpon þe next figur{es} hede afor{e} toward þe ryȝt side, þe quych is now 4. but afor{e} it was 8. for þat 1 schal be rekenet for 10, of þe quych 10, 5 is halfe, as þou knowest wel. Whan þ{o}u hast þus ydo, medye þ{a}t þe quych leues aft{er} þe takying{e} away of þat þat is odde, þe quych leuyng{e} schall{e} be 3; do away 6 & sette þ{er}e 3, & þou schalt haue such a nombre

5
4634.

aft{er} go forth to þe next fig{ur}e, & medy þat, & worch forth, for it is lyȝt ynovȝt to þe c{er}tayɳ.

¶ Si v{er}o s{e}c{un}da dat vnu{m}.
Illa deleta, sc{ri}bat{ur} cifra; p{ri}ori
¶ Tradendo quinque pro denario mediato;
Nec cifra sc{ri}batur, nisi dei{n}de fig{ur}a seq{u}at{ur}:
Postea p{ro}cedas reliq{ua}s mediando figuras
Vt sup{ra} docui, si sint tibi mille figure.

[Sidenote: If the second figure is one, put a cipher, and write
five over the next figure. How to halve fourteen.]

¶ Her{e} he putt{es} þe 5 case, þe quych is [*leaf 151a] þis: yf þe secunde figur{e} be of 1, as þis is here 12, þou schalt do away þat 1 & sett þ{ere} a cifer. & sett 5 ou{er} þe next fig{ur}e hede afor{e} toward þe riȝt side, as þou diddyst afor{e}; & þat 5 schal be haldel of þat 1, þe quych 1 is rekent for 10. lo an Ensampull{e}, 214. medye 4. þ{a}t schall{e} be 2. do away 4 & sett þ{ere} 2. þe{n} go forth to þe next figur{e}. þe quych is bot 1. do away þat 1. & sett þ{ere} a cifer. & set 5 vpon þe figur{es} hed afor{e}, þe quych is nowe 2, & þen þou schalt haue þis no{m}b{re}

5
202,

þen worch forth to þe nex fig{ur}e. And also it is no mayst{er}y yf þ{ere} come no figur{e} after þat on is medyet, þ{o}u schalt write no 0. ne nowȝt ellis, but set 5 ou{er} þe next fig{ur}e afor{e} toward þe ryȝt, as þus 14. medie 4 then leues 2, do away 4 & sett þ{ere} 2. þen medie 1. þe q{ui}ch is rekende for ten, þe halue{n}del þ{ere}-of wel be 5. sett þ{a}t 5 vpon þe hede of þ{a}t figur{e}, þe quych is now 2, & do away þ{a}t 1, & þou schalt haue þis nombre yf þ{o}u worch wel,

5
2.

vn{de} v{er}sus.

[Headnote: How to prove the Mediation.]

¶ Si mediacio sit b{e}n{e} f{ac}ta p{ro}bar{e} valeb{is}
¶ Duplando num{er}u{m} que{m} p{ri}mo di{m}ediasti

[Sidenote: How to prove your mediation. First example. The second.
The third example. The fourth example. The fifth example.]

¶ Her{e} he telles þe how þou schalt know wheþ{er} þou hase wel ydo or no. doubul [*leaf 151b] þe nombre þe quych þ{o}u hase mediet, and yf þ{o}u haue wel y-medyt after þe dupleacioɳ, þou schalt haue þe same nombre þat þ{o}u haddyst in þe tabull{e} or þ{o}u began to medye, as þus. ¶ The furst ensampull{e} was þis. 4. þe quych I-mediet was laft 2, þe whych 2 was write in þe place þ{a}t 4 was write afor{e}. Now doubull{e} þat 2, & þ{o}u schal haue 4, as þ{o}u hadyst afor{e}. þe secunde Ensampull{e} was þis, 245. When þ{o}u haddyst mediet all{e} þis nomb{re}, yf þou haue wel ydo þou schalt haue of þ{a}t mediacioɳ þis nombre, 122ʷ. Now doubull{e} þis nombre, & begyn in þe lyft side; doubull{e} 1, þat schal be 2. do away þat 1 & sett þ{ere} 2. þen doubull{e} þ{a}t oþ{er} 2 & sett þ{ere} 4, þen doubull{e} þat oþ{er} 2, & þat wel be 4. þe{n} doubul þat merke þat stondes for halue on. & þat schall{e} be 1. Cast þat on to 4, & it schall{e} be 5. do away þat 2 & þat merke, & sette þ{ere} 5, & þen þ{o}u schal haue þis nombre 245. & þis wos þe same nombur þ{a}t þ{o}u haddyst or þ{o}u began to medye, as þ{o}u mayst se yf þou take hede. The nombre þe quych þou haddist for an Ensampul in þe 3 case of mediacioɳ to be mediet was þis 241. whan þ{o}u haddist medied all{e} þis nombur truly [*leaf 152a] by eu{er}y figur{e}, þou schall haue be þ{a}t mediacioɳ þis nombur 120ʷ. Now dowbul þis nomb{ur}, & begyn in þe lyft side, as I tolde þe in þe Craft of duplacioɳ. þus doubull{e} þe fig{ur}e of 1, þat wel be 2. do away þat 1 & sett þ{ere} 2, þen doubul þe next figur{e} afore, the quych is 2, & þat wel be 4; do away 2 & set þ{ere} 4. þen doubul þe cifer, & þat wel be noȝt, for a 0 is noȝt. And twyes noȝt is but noȝt. þ{ere}for{e} doubul the merke aboue þe cifers hede, þe quych betokenes þe halue{n}del of 1, & þat schal be 1. do away þe cifer & þe merke, & sett þ{ere} 1, & þen þ{o}u schalt haue þis nombur 241. And þis same nombur þ{o}u haddyst afore or þ{o}u began to medy, & yf þ{o}u take gode hede. ¶ The next ensampul þat had in þe 4 case of mediacioɳ was þis 4678. Whan þ{o}u hast truly ymedit all{e} þis nombur fro þe begynnyng{e} to þe endyng{e}, þ{o}u schalt haue of þe mediacioɳ þis nombur

5
2334.

Now doubul this nombur & begyn in þe lyft side, & doubull{e} 2 þat schal be 4. do away 2 and sette þere 4; þen doubul{e} 3, þ{a}t wol be 6; do away 3 & sett þ{ere} 6, þen doubul þat oþ{er} 3, & þat wel be 6; do away 3 & set þ{ere} [*leaf 152b] 6, þen doubul þe 4, þat welle be 8; þen doubul 5. þe quych stondes ou{er} þe hed of 4, & þat wol be 10; cast 10 to 8, & þ{a}t schal be 18; do away 4 & þat 5, & sett þ{ere} 8, & sett that 1, þe quych is an articul of þe Composit þe quych is 18, ou{er} þe next figur{es} hed toward þe lyft side, þe quych is 6. drav þ{a}t 1 to 6, þe quych 1 in þe dravyng schal be rekente bot for 1, & þ{a}t 1 & þ{a}t 6 togedur wel be 7. do away þat 6 & þat 1. the quych stondes ou{er} his hede, & sett ther 7, & þen þou schalt haue þis nombur 4678. And þis same nombur þ{o}u hadyst or þ{o}u began to medye, as þ{o}u mayst see in þe secunde Ensampul þat þou had in þe 4 case of mediacioɳ, þat was þis: when þ{o}u had mediet truly all{e} the nombur, a p{ri}ncipio usque ad fine{m}. þ{o}u schalt haue of þat mediacioɳ þis nombur

5
102.

Now doubul 1. þat wel be 2. do away 1 & sett þ{ere} 2. þen doubul 0. þ{a}t will be noȝt. þ{ere}for{e} take þe 5, þe quych stondes ou{er} þe next figur{es} hed, & doubul it, & þat wol be 10. do away þe 0 þat stondes betwene þe two fig{u}r{i}s, & sette þ{ere} in his stid 1, for þ{a}t 1 now schal stonde in þe secunde place, wher{e} he schal betoken 10; þen doubul 2, þat wol be 4. do away 2 & sett þere 4. & [*leaf 153a] þou schal haue þus nombur 214. þis is þe same nu{m}bur þat þ{o}u hadyst or þ{o}u began to medye, as þ{o}u may see. And so do eu{er} mor{e}, yf þ{o}u wil knowe wheþ{er} þou hase wel ymedyt or no. ¶. doubull{e} þe nu{m}bur þat comes aft{er} þe mediaciouɳ, & þ{o}u schal haue þe same nombur þ{a}t þ{o}u hadyst or þ{o}u began to medye, yf þ{o}u haue welle ydo. or els doute þe noȝt, but yf þ{o}u haue þe same, þ{o}u hase faylide in þ{i} Craft.

+Sequitur de multiplicatione.+

[Headnote: The Craft of Multiplication.]

[Headnote: To write down a Multiplication Sum.]

++Si tu p{er} num{er}u{m} num{er}u{m} vis m{u}ltiplicar{e}
Scribe duas q{ua}scu{nque} velis series nu{me}ror{um}
Ordo s{er}vet{ur} vt vltima m{u}ltiplicandi
Ponat{ur} sup{er} ant{er}iorem multiplicant{is}
A leua reliq{u}e sint scripte m{u}ltiplicantes.

[Sidenote: Four things to be known of Multiplication: the first:
the second: the third: the fourth. How to set down the sum. Two
sorts of Multiplication: mentally, and on paper.]

¶ Her{e} be-gynnes þe Chapt{r}e of m{u}ltiplicatioɳ, in þe quych þou most know 4 thynges. ¶ Ffirst, qwat is m{u}ltiplicacioɳ. The secunde, how mony cases may hap in multiplicacioɳ. The thryde, how mony rewes of figur{es} þ{ere} most be. ¶ The 4. what is þe p{ro}fet of þis craft. ¶ As for þe first, þ{o}u schal vnd{er}stonde þat m{u}ltiplicacioɳ is a bryngyng{e} to-ged{er} of 2 thyng{es} in on nombur, þe quych on nombur {con}tynes so mony tymes on, howe [*leaf 153b] mony tymes þ{ere} ben vnytees in þe nowmb{re} of þat 2, as twyes 4 is 8. now her{e} ben þe 2 nomb{er}s, of þe quych too nowmbr{e}s on is betokened be an adu{er}be, þe quych is þe worde twyes, & þis worde thryes, & þis worde four{e} sythes,[{9}] [[& þis wordes fyue sithe & sex sythes.]] & so furth of such other lyke wordes. ¶ And tweyn nombres schal be tokenyde be a nowne, as þis worde four{e} showys þes tweyɳ nombres y-broth in-to on hole nombur, þat is 8, for twyes 4 is 8, as þ{o}u wost wel. ¶ And þes nomb{re} 8 conteynes as oft tymes 4 as þ{ere} ben vnites in þ{a}t other nomb{re}, þe quych is 2, for in 2 ben 2 vnites, & so oft tymes 4 ben in 8, as þ{o}u wottys wel. ¶ ffor þe secu{n}de, þ{o}u most know þat þ{o}u most haue too rewes of figures. ¶ As for þe thryde, þ{o}u most know þ{a}t 8 man{er} of diu{er}se case may happe in þis craft. The p{ro}fet of þis Craft is to telle when a nomb{re} is m{u}ltiplyed be a noþ{er}, qwat co{m}mys þ{ere} of. ¶ fforthermor{e}, as to þe sentence of our{e} verse, yf þ{o}u wel m{u}ltiply a nombur be a-noþ{er} nomb{ur}, þou schalt write [*leaf 154a] a rewe of figures of what nomb{ur}s so eu{er} þ{o}u welt, & þat schal be called Num{erus} m{u}ltiplicand{us}, Anglice, þe nomb{ur} the quych to be m{u}ltiplied. þen þ{o}u schalt write a-nother rewe of figur{e}s, by þe quych þ{o}u schalt m{u}ltiplie the nombre þat is to be m{u}ltiplied, of þe quych nomb{ur} þe furst fig{ur}e schal be write vnd{er} þe last figur{e} of þe nomb{ur}, þe quych is to be m{u}ltiplied. And so write forthe toward þe lyft side, as her{e} you may se,

+----------+
| 67324 |
| 1234 |
+----------+

And þis on{e} nomb{ur} schall{e} be called nu{meru}s m{u}ltiplicans. An{gli}ce, þe nomb{ur} m{u}ltipliyng{e}, for he schall{e} m{u}ltiply þe hyer nounb{ur}, as þus on{e} tyme 6. And so forth, as I schal telle the aft{er}warde. And þou schal begyn in þe lyft side. ¶ ffor-þ{ere}-more þou schalt vndurstonde þat þ{ere} is two man{ur}s of m{u}ltiplicacioɳ; one ys of þe wyrchyng{e} of þe boke only in þe mynde of a mon. fyrst he teches of þe fyrst man{er} of duplacioɳ, þe quych is be wyrchyng{e} of tabuls. Aft{er}warde he wol teche on þe secunde man{er}. vn{de} v{er}sus.

[Headnote: To multiply one Digit by another.]

In digitu{m} cures digitu{m} si duc{er}e ma{i}or
[*leaf 154b.]
P{er} qua{n}tu{m} distat a denis respice debes
¶ Namq{ue} suo decuplo totiens deler{e} mi{n}ore{m}
Sitq{ue} tibi nu{meru}s veniens exinde patebit.

[Sidenote: How to multiply two digits. Subtract the greater from ten;
take the less so many times from ten times itself. Example.]

¶ Her{e} he teches a rewle, how þ{o}u schalt fynde þe nounb{r}e þat comes by þe m{u}ltiplicacioɳ of a digit be anoþ{er}. loke how mony [vny]tes ben. bytwene þe mor{e} digit and 10. And reken ten for on vnite. And so oft do away þe lasse nounbre out of his owne decuple, þat is to say, fro þat nounb{r}e þat is ten tymes so mych is þe nounb{re} þ{a}t comes of þe m{u}ltiplicacioɳ. As yf þ{o}u wol m{u}ltiply 2 be 4. loke how mony vnitees ben by-twene þe quych is þe mor{e} nounb{re}, & be-twene ten. C{er}ten þ{ere} wel be vj vnitees by-twene 4 & ten. yf þ{o}u reken þ{ere} w{i}t{h} þe ten þe vnite, as þou may se. so mony tymes take 2. out of his decuple, þe quych is 20. for 20 is þe decuple of 2, 10 is þe decuple of 1, 30 is þe decuple of 3, 40 is þe decuple of 4, And þe oþ{er} digetes til þ{o}u come to ten; & whan þ{o}u hast y-take so mony tymes 2 out of twenty, þe quych is sex tymes, þ{o}u schal leue 8 as þ{o}u wost wel, for 6 times 2 is twelue. take [1]2 out of twenty, & þ{ere} schal leue 8. bot yf bothe þe digett{es} [*leaf 155a] ben y-lyech mych as her{e}. 222 or too tymes twenty, þen it is no fors quych of hem tweyn þ{o}u take out of here decuple. als mony tymes as þ{a}t is fro 10. but neu{er}-þe-lesse, yf þ{o}u haue hast to worch, þ{o}u schalt haue her{e} a tabul of figures, wher{e}-by þ{o}u schalt se a-nonɳ ryght what is þe nounbre þ{a}t comes of þe multiplicacioɳ of 2 digittes. þus þ{o}u schalt worch in þis fig{ur}e.

[Sidenote: Better use this table, though. How to use it. The way to
use the Multiplication table.]

1|
-----
2| 4|
--------
3| 6| 9|
-----------
4| 8|12|16|
--------------
5|10|15|20|25|
-----------------
6|12|18|24|30|36|
--------------------
7|14|21|28|35|42|49|
-----------------------
8|16|24|32|40|48|56|64|
--------------------------
9|18|27|36|45|54|63|72|81|
----------------------------
1| 2| 3| 4| 5| 6| 7| 8| 9|
----------------------------

yf þe fig{ur}e, þe quych schall{e} be m{u}ltiplied, be euen{e} as mych as þe diget be, þe quych þat oþ{er} figur{e} schal be m{u}ltiplied, as two tymes twayɳ, or thre tymes 3. or sych other. loke qwer{e} þat fig{ur}e sittes in þe lyft side of þe t{ri}angle, & loke qwer{e} þe diget sittes in þe neþ{er} most rewe of þe triangle. & go fro hym vpwarde in þe same rewe, þe quych rewe gose vpwarde til þ{o}u come agaynes þe oþ{er} digette þat sittes in þe lyft side of þe t{ri}angle. And þat nounbre, þe quych þou [*leaf 155b] fyn[*]des þ{ere} is þe nounbre þat comes of the m{u}ltiplicacioɳ of þe 2 digittes, as yf þou wold wete qwat is 2 tymes 2. loke quer{e} sittes 2 in þe lyft side i{n} þe first rewe, he sittes next 1 in þe lyft side al on hye, as þ{o}u may se; þe[{n}] loke qwer{e} sittes 2 in þe lowyst rewe of þe t{ri}angle, & go fro hym vpwarde in þe same rewe tyll{e} þou come a-ȝenenes 2 in þe hyer place, & þer þou schalt fynd ywrite 4, & þat is þe nounb{r}e þat comes of þe multiplicacioɳ of two tymes tweyn is 4, as þow wotest well{e}. yf þe diget. the quych is m{u}ltiplied, be mor{e} þan þe oþ{er}, þou schalt loke qwer{e} þe mor{e} diget sittes in þe lowest rewe of þe t{ri}angle, & go vpwarde in þe same rewe tyl[{10}] þ{o}u come a-nendes þe lasse diget in the lyft side. And þ{ere} þ{o}u schalt fynde þe no{m}b{r}e þat comes of þe m{u}ltiplicacioɳ; but þ{o}u schalt vnd{er}stonde þat þis rewle, þe quych is in þis v{er}se. ¶ In digitu{m} cures, &c., noþ{er} þis t{ri}angle schall{e} not s{er}ue, bot to fynde þe nounbres þ{a}t comes of the m{u}ltiplicacioɳ þat comes of 2 articuls or {com}posites, þe nedes no craft but yf þou wolt m{u}ltiply in þi mynde. And [*leaf 156a] þere-to þou schalt haue a craft aft{er}warde, for þou schall wyrch w{i}t{h} digettes in þe tables, as þou schalt know aft{er}warde. v{er}sus.

[Headnote: To multiply one Composite by another.]

¶ Postea p{ro}cedas postrema{m} m{u}ltiplica{n}do
[Recte multiplicans per cu{n}ctas i{n}feriores]
Condic{i}onem tamen t{a}li q{uod} m{u}ltiplicant{es}
Scribas in capite quicq{ui}d p{ro}cesserit inde
Sed postq{uam} fuit hec m{u}ltiplicate fig{ur}e
Anteriorent{ur} serei m{u}ltiplica{n}t{is}
Et sic m{u}ltiplica velut isti m{u}ltiplicasti
Qui sequit{ur} nu{mer}u{m} sc{ri}ptu{m} quiscu{n}q{ue} figur{is}.

[Sidenote: How to multiply one number by another. Multiply the ‘last’
figure of the higher by the ‘first’ of the lower number. Set the
answer over the first of the lower: then multiply the second of the
lower, and so on. Then antery the lower number: as thus. Now multiply
by the last but one of the higher: as thus. Antery the figures again,
and multiply by five: Then add all the figures above the line: and
you will have the answer.]

¶ Her{e} he teches how þ{o}u schalt wyrch in þis craft. þou schalt m{ul}tiplye þe last figur{e} of þe nombre, and quen þ{o}u hast so ydo þou schalt draw all{e} þe figures of þe neþ{er} nounbre mor{e} taward þe ryȝt side, so qwe{n} þ{o}u hast m{u}ltiplyed þe last figur{e} of þe heyer nounbre by all{e} þe neþ{er} figures. And sette þe nounbir þat comes þer-of ou{er} þe last figur{e} of þe neþ{er} nounb{re}, & þen þou schalt sette al þe oþ{er} fig{ur}es of þe neþ{er} nounb{re} mor{e} ner{e} to þe ryȝt side. ¶ And whan þou hast m{u}ltiplied þat figur{e} þat schal be m{u}ltiplied þe next aft{er} hym by al þe neþ{er} figures. And worch as þou dyddyst afor{e} til [*leaf 156b] þou come to þe ende. And þou schalt vnd{er}stonde þat eu{er}y figur{e} of þe hier nounb{re} schal be m{u}ltiplied be all{e} þe figur{e}s of the neþ{er} nounbre, yf þe hier nounb{re} be any figur{e} þen on{e}. lo an Ensampul her{e} folowyng{e}.

+------+
| 2465|.
|232 |
+------+

þou schalt begyne to m{u}ltiplye in þe lyft side. M{u}ltiply 2 be 2, and twyes 2 is 4. set 4 ou{er} þe hed of þ{a}t 2, þen m{u}ltiplie þe same hier 2 by 3 of þe nether nounbre, as thryes 2 þat schal be 6. set 6 ou{er} þe hed of 3, þan m{u}ltiplie þe same hier 2 by þat 2 þe quych stondes vnd{er} hym, þ{a}t wol be 4; do away þe hier 2 & sette þ{ere} 4. ¶ Now þ{o}u most antery þe nether nounbre, þat is to say, þ{o}u most sett þe neþ{er} nounbre more towarde þe ryȝt side, as þus. Take þe neþ{er} 2 toward þe ryȝt side, & sette it eueɳ vnd{er} þe 4 of þe hyer nounb{r}e, & ant{er}y all{e} þe figures þat comes aft{er} þat 2, as þus; sette 2 vnd{er} þe 4. þen sett þe figur{e} of 3 þ{ere} þat þe figure of 2 stode, þe quych is now vndur þ{a}t 4 in þe hier nounbre; þen sett þe oþer figur{e} of 2, þe quych is þe last fig{ur}e toward þe lyft side of þe neþ{er} nomb{er} þ{ere} þe figur{e} of 3 stode. þen þ{o}u schalt haue such a nombre.

+------+
|464465|
| 232 |
+------+

[*leaf 157a] ¶ Now m{u}ltiply 4, þe quych comes next aft{er} 6, by þe last 2 of þe neþ{er} nounbur toward þe lyft side. as 2 tymes 4, þat wel be 8. sette þat 8 ou{er} þe figure the quych stondes ou{er} þe hede of þat 2, þe quych is þe last figur{e} of þe neþ{er} nounbre; þan multiplie þat same 4 by 3, þat comes in þe neþ{er} rewe, þat wol be 12. sette þe digit of þe composyt ou{er} þe figure þe quych stondes ou{er} þe hed of þat 3, & sette þe articule of þis co{m}posit ou{er} al þe figures þat stondes ou{er} þe neþ{er} 2 hede. þen m{u}ltiplie þe same 4 by þe 2 in þe ryȝt side in þe neþ{er} nounbur, þat wol be 8. do away 4. & sette þ{ere} 8. Eu{er} mor{e} qwen þ{o}u m{u}ltiplies þe hier figur{e} by þat figur{e} þe quych stondes vnd{er} hym, þou schalt do away þat hier figur{e}, & sett þer þat nounbre þe quych comes of m{u}ltiplicacioɳ of ylke digittes. Whan þou hast done as I haue byde þe, þ{o}u schalt haue suych an ord{er} of figur{e} as is her{e},

+--------+
| 1 |.
| 82 |
|4648[65]|
| 232 |
+--------+

þen take and ant{er}y þi neþ{er} figures. And sett þe fyrst fig{ur}e of þe neþ{er} figures[{11}] vndre be figur{e} of 6. ¶ And draw al þe oþ{er} figures of þe same rewe to hym-warde, [*leaf 157b] as þ{o}u diddyst afore. þen m{u}ltiplye 6 be 2, & sett þat þe quych comes ou{er} þ{ere}-of ou{er} al þe oþ{er} figures hedes þat stondes ou{er} þat 2. þen m{u}ltiply 6 be 3, & sett all{e} þat comes þ{ere}-of vpon all{e} þe figur{e}s hedes þat standes ou{er} þat 3; þa{n} m{u}ltiplye 6 be 2, þe quych stondes vnd{er} þat 6, þen do away 6 & write þ{ere} þe digitt of þe composit þat schal come þ{ere}of, & sette þe articull ou{er} all{e} þe figures þat stondes ou{er} þe hede of þat 3 as her{e},

+------+
| 11 |
| 121 |
| 828 |
|464825|
| 232 |
+------+

þen ant{er}y þi figures as þou diddyst afor{e}, and m{u}ltipli 5 be 2, þat wol be 10; sett þe 0 ou{er} all þe figures þ{a}t stonden ou{er} þat 2, & sett þ{a}t 1. ou{er} the next figures hedes, all{e} on hye towarde þe lyft side. þen m{u}ltiplye 5 be 3. þat wol be 15, write 5 ou{er} þe figures hedes þat stonden ou{er} þ{a}t 3, & sett þat 1 ou{er} þe next figur{e}s hedes toward þe lyft side. þen m{u}ltiplye 5 be 2, þat wol be 10. do away þat 5 & sett þ{ere} a 0, & sett þat 1 ou{er} þe figures hedes þat stonden ou{er} 3. And þen þou schalt haue such a nounbre as here stondes aftur.[*leaf 158a]

+------+
| 11 |
| 1101 |
| 1215 |
| 82820|
|4648 |
| 232|
+------+

¶ Now draw all{e} þese figures downe toged{er} as þus, 6.8.1. & 1 draw to-gedur; þat wolle be 16, do away all{e} þese figures saue 6. lat hym stonde, for þow þ{o}u take hym away þou most write þer þe same aȝene. þ{ere}for{e} late hym stonde, & sett 1 ou{er} þe figur{e} hede of 4 toward þe lyft side; þen draw on to 4, þat woll{e} be 5. do away þat 4 & þat 1, & sette þ{ere} 5. þen draw 4221 & 1 toged{ur}, þat wol be 10. do away all{e} þat, & write þere þat 4 & þat 0, & sett þat 1 ou{er} þe next figur{es} hede toward þe lyft side, þe quych is 6. þen draw þat 6 & þat 1 togedur, & þat wolle be 7; do away 6 & sett þ{ere} 7, þen draw 8810 & 1, & þat wel be 18; do away all{e} þe figures þ{a}t stondes ou{er} þe hede of þat 8, & lette 8 stonde stil, & write þat 1 ou{er} þe next fig{u}r{is} hede, þe quych is a 0. þen do away þat 0, & sett þ{ere} 1, þe quych stondes ou{er} þe 0. hede. þen draw 2, 5, & 1 toged{ur}, þat woll{e} be 8. þen do away all{e} þat, & write þ{ere} 8. ¶ And þen þou schalt haue þis nounbre, 571880.

[Headnote: The Cases of this Craft.]

[*leaf 158b]

¶ S{ed} cu{m} m{u}ltiplicabis, p{ri}mo sic e{st} op{er}andu{m},
Si dabit articulu{m} tibi m{u}ltiplicacio solu{m};
P{ro}posita cifra su{m}ma{m} t{ra}nsferre meme{n}to.

[Sidenote: What to do if the first multiplication results in an
article.]

¶ Her{e} he puttes þe fyrst case of þis craft, þe quych is þis: yf þ{ere} come an articulle of þe m{u}ltiplicacioɳ ysette befor{e} the articull{e} in þe lyft side as þus

+---+
| 51|.
|23 |
+---+

multiplye 5 by 2, þat wol be 10; sette ou{er} þe hede of þat 2 a 0, & sett þat on, þat is þe articul, in þe lyft side, þat is next hym, þen þ{o}u schalt haue þis nounbre

+----+
|1051|.
| 23 |
+----+

¶ And þen worch forth as þou diddist afore. And þ{o}u schalt vnd{er}stonde þat þ{o}u schalt write no 0. but whan þat place where þou schal write þat 0 has no figure afore hy{m} noþ{er} aft{er}. v{er}sus.

¶ Si aut{em} digitus excreu{er}it articul{us}q{ue}.
Articul{us}[{12}] sup{ra}p{osit}o digito salit vltra.

[Sidenote: What to do if the result is a composite number.]

¶ Her{e} is þe secunde case, þe quych is þis: yf hit happe þat þ{ere} come a composyt, þou schalt write þe digitte ou{er} þe hede of þe neþ{er} figur{e} by þe quych þ{o}u multipliest þe hier figure; and sett þe articull{e} next hym toward þe lyft side, as þou diddyst afore, as þ{us}

+---+
| 83|.
|83 |
+---+